[PPT] - On Distributionally Robust Chance Constrained Program with PowerPoint Presentation

SLIDE 1

On Distributionally Robust Chance Constrained Program with Wasserstein Distance

Weijun Xie ISE, Virginia Tech Mathematical Optimization of Systems Impacted by Rare, High-Impact Random Events, Jun 24 - 28, 2019

SLIDE 2

Distributionally Robust Chance Constrained Program (DRCCP)

Consider DRCCP as v∗ = min

x

c⊤x (objective function) s.t. x ∈ S (deterministic constraints) e.g., nonnegativity ˜ Ax ≥ ˜ b (uncertain inequalities)

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 2 / 27

SLIDE 3

Distributionally Robust Chance Constrained Program (DRCCP)

Consider DRCCP as v∗ = min

x

c⊤x (objective function) s.t. x ∈ S (deterministic constraints) e.g., nonnegativity inf

P∈P P{ ˜

Ax ≥ ˜ b} ≥ 1 − ǫ (chance constraint) where

◮ ǫ ∈ (0, 1) is risk parameter ◮ “Ambiguity Set” P = a family of probability distributions

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 2 / 27

SLIDE 4

Distributionally Robust Chance Constrained Program (DRCCP)

Consider DRCCP as v∗ = min

x

c⊤x (objective function) s.t. x ∈ S (deterministic constraints) e.g., nonnegativity inf

P∈P P

     ˜ a⊤

1 x ≥ ˜

b1 . . . ˜ a⊤

mx ≥ ˜

bm      ≥ 1 − ǫ (chance constraint) where

◮ ǫ ∈ (0, 1) is risk parameter ◮ “Ambiguity Set” P = a family of probability distributions ◮ m = 1: single DRCCP; m > 1: joint DRCCP

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 2 / 27

SLIDE 5

Wasserstein Ambiguity Set

Wasserstein ambiguity set (Esfahani and Kuhn 2015; Zhao and Guan, 2015; Gao and Kleywegt, 2016; Blanchet and Murthy, 2016) PW =

P : Wq
P, P˜

ζ

≤ δ
,

where Wq

P, P˜

ζ

= Wasserstein distance between probability distribution P

and empirical distribution P˜

ζ.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 3 / 27

SLIDE 6

Wasserstein Ambiguity Set

Wasserstein ambiguity set (Esfahani and Kuhn 2015; Zhao and Guan, 2015; Gao and Kleywegt, 2016; Blanchet and Murthy, 2016) PW =

P : Wq
P, P˜

ζ

≤ δ
,

where Wq

P, P˜

ζ

= Wasserstein distance between probability distribution P

and empirical distribution P˜

ζ.

◮ Convergence in probability to regular chance constrained program (CCP) ◮

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 3 / 27

SLIDE 7

DRCCP with Wasserstein Ambiguity Set (DRCCP-W): Existing Works

DRCCP-W set Z =

x : inf

P∈PW P

˜ Ax ≥ ˜ b

≥ 1 − ǫ
,

with PW =

P : Wq
P, P˜

ζ

≤ δ
.

◮ Hanasusanto et al. (2015) and X. and Ahmed (2017) showed that

DRCCP-W is a biconvex program.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 4 / 27

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DRCCP with Wasserstein Ambiguity Set (DRCCP-W): Existing Works

DRCCP-W set Z =

x : inf

P∈PW P

˜ Ax ≥ ˜ b

≥ 1 − ǫ
,

with PW =

P : Wq
P, P˜

ζ

≤ δ
.

◮ Hanasusanto et al. (2015) and X. and Ahmed (2017) showed that

DRCCP-W is a biconvex program.

◮ X. and Ahmed (2017) proposed a bicriteria approximation algorithm for

a special family of DRCCP-W

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 4 / 27

SLIDE 9

DRCCP-W: Summary of Contributions

DRCCP-W set Z =

x : inf

P∈PW P

˜ Ax ≥ ˜ b

≥ 1 − ǫ
.

◮ DRCCP-W ≡ conditional-value-at-risk (CVaR) constrained optimization

Develop inner and outer approximations

◮ DRCCP-W set Z is mixed integer program representable

With big-M coefficients and additional binary variables

◮ Binary DRCCP-W set (i.e., S ⊆ {0, 1}n) is submodular constrained

Without big-M coefficients and additional binary variables Solvable by Branch and Cut Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 5 / 27

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Outline

◮ CVaR Reformulation and Related Approximations ◮ Mixed Integer Program Reformulation ◮ Binary DRCCP-W and Submodularity ◮ Concluding Remarks

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 6 / 27

SLIDE 11

CVaR Reformulation and Related Approximations

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 7 / 27

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CVaR Reformulation

DRCCP-W set Z =

x : inf

P∈PW P

˜ Ax ≥ ˜ b

≥ 1 − ǫ
,

with PW =

P : Wq
P, P˜

ζ

≤ δ
.

Theorem (Exact Formulation)

Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

inf

a⊤

i x<bi

(ai, bi) − (aζ

i , bζ i ) and

CVaR1−ǫ

˜

X

= min

γ

γ + 1

ǫ EP

˜

X − γ

+
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 8 / 27

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CVaR Reformulation

DRCCP-W set Z =

x : inf

P∈PW P

˜ Ax ≥ ˜ b

≥ 1 − ǫ
,

with PW =

P : Wq
P, P˜

ζ

≤ δ
.

Theorem (Exact Formulation)

Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

inf

a⊤

i x<bi

(ai, bi) − (aζ

i , bζ i ) and

CVaR1−ǫ

˜

X

= min

γ

γ + 1

ǫ EP

˜

X − γ

+
.

Proof Idea: (1) strong duality of distributionally robust optimization, and (2) break down the indicator function.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 8 / 27

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CVaR Reformulation: Worst-case Interpretation

Theorem (Exact Formulation)

Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

inf

a⊤

i x<bi

(ai, bi) − (aζ

i , bζ i )

Original empirical samples

◮ N = 6, ǫ = 1/3

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 9 / 27

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CVaR Reformulation: Worst-case Interpretation

Theorem (Exact Formulation)

Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

inf

a⊤

i x<bi

(ai, bi) − (aζ

i , bζ i )

Original empirical samples Moving these samples to boundary of violating constraints

◮ N = 6, ǫ = 1/3

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 9 / 27

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CVaR Reformulation: Worst-case Interpretation

Theorem (Exact Formulation)

Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

inf

a⊤

i x<bi

(ai, bi) − (aζ

i , bζ i )

Original empirical samples Moving these samples to boundary of violating constraints Due to chance constraint,

nly limited scenarios can

be moved

◮ N = 6, ǫ = 1/3

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 9 / 27

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CVaR Reformulation: Simplification

DRCCP-W set Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

inf

a⊤

i x<bi

(ai, bi) − (aζ

i , bζ i ).

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27

SLIDE 18

CVaR Reformulation: Simplification

DRCCP-W set Z =

x : δ

ǫ + CVaR1−ǫ

−f(x, ˜

ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

max

(aζ

i )⊤x − bζ i , 0

(x, 1)∗

.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27

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CVaR Reformulation: Simplification

DRCCP-W set Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m] max

(aζ

i )⊤x − bζ i , 0

.

◮ By positive homogeneity of coherent risk measures

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27

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CVaR Reformulation: Simplification

DRCCP-W set Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

◮ By positive homogeneity of coherent risk measures ◮ Switch minimax to maximin

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27

SLIDE 21

Outer Approximation

Note CVaR1−ǫ

˜

X

≥ VaR1−ǫ
˜

X

:= min
s : F ˜

X(s) ≥ 1 − ǫ

.

Replace CVaR1−ǫ

˜

X

by VaR1−ǫ
˜

X

.

Theorem (Outer Approximation)

Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
where

f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 11 / 27

SLIDE 22

Outer Approximation

Note CVaR1−ǫ

˜

X

≥ VaR1−ǫ
˜

X

:= min
s : F ˜

X(s) ≥ 1 − ǫ

.

Replace CVaR1−ǫ

˜

X

by VaR1−ǫ
˜

X

.

Theorem (Outer Approximation)

Z ⊆ ZVaR =

x : δ

ǫ(x, 1)∗ + VaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
where

f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 11 / 27

SLIDE 23

Outer Approximation

Note CVaR1−ǫ

˜

X

≥ VaR1−ǫ
˜

X

:= min
s : F ˜

X(s) ≥ 1 − ǫ

.

Replace CVaR1−ǫ

˜

X

by VaR1−ǫ
˜

X

.

Theorem (Outer Approximation)

Z ⊆ ZVaR =

x : P˜

ζ

˜

Aζx ≥ ˜ bζ + δ ǫ(x, −1)∗e

≥ 1 − ǫ
Xie (Virginia Tech)

DRCCP with Wasserstein Distance June 26, 2019 11 / 27

SLIDE 24

Outer Approximation

Note CVaR1−ǫ

˜

X

≥ VaR1−ǫ
˜

X

:= min
s : F ˜

X(s) ≥ 1 − ǫ

.

Replace CVaR1−ǫ

˜

X

by VaR1−ǫ
˜

X

.

Theorem (Outer Approximation)

Z ⊆ ZVaR =

x : P˜

ζ

˜

Aζx ≥ ˜ bζ + δ ǫ(x, −1)∗e

≥ 1 − ǫ
Remarks.

◮ Asymptotically optimal, i.e., ZVaR → Z as δ → 0+ ◮ Regular CCP: many existing methods

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 11 / 27

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Inner Approximation: Scenario Approach

Note CVaR1−ǫ

˜

X

≤ CVaR1
˜

X

:= ess. sup( ˜

X). Replace CVaR1−ǫ

˜

X

by ess. sup( ˜

X).

Theorem (Inner Approximation)

Z =

x : δ

ǫ(x, −1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
where

f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 12 / 27

SLIDE 26

Inner Approximation: Scenario Approach

Note CVaR1−ǫ

˜

X

≤ CVaR1
˜

X

:= ess. sup( ˜

X). Replace CVaR1−ǫ

˜

X

by ess. sup( ˜

X).

Theorem (Inner Approximation)

Z ⊇ ZS =

x : δ

ǫ(x, −1)∗ + ess. sup

−

f(x, ˜ ζ)

≤ 0
where

f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 12 / 27

SLIDE 27

Inner Approximation: Scenario Approach

Note CVaR1−ǫ

˜

X

≤ CVaR1
˜

X

:= ess. sup( ˜

X). Replace CVaR1−ǫ

˜

X

by ess. sup( ˜

X).

Theorem (Inner Approximation)

Z ⊇ ZS =

x ∈ Rn : P˜

ζ

˜

Aζx ≥ ˜ bζ + δ ǫ(x, −1)∗e

= 1
Xie (Virginia Tech)

DRCCP with Wasserstein Distance June 26, 2019 12 / 27

SLIDE 28

Inner Approximation: Scenario Approach

Note CVaR1−ǫ

˜

X

≤ CVaR1
˜

X

:= ess. sup( ˜

X). Replace CVaR1−ǫ

˜

X

by ess. sup( ˜

X). Suppose ˜ ζ has finite support {(Aζ, bζ)}ζ∈[N].

Theorem (Inner Approximation)

Z ⊇ ZS =

x ∈ Rn : Aζx ≥ bζ + δ

ǫ(x, −1)∗e, ∀ζ ∈ [N]

Remarks.

◮ ZS is a conic set ◮ ZS ≡ the robust scenario approach (Calafiore and Campi, 2006) to

regular CCP when the sample size is small

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 12 / 27

SLIDE 29

Inner Approximation: Scenario Approach

Note CVaR1−ǫ

˜

X

≤ CVaR1
˜

X

:= ess. sup( ˜

X). Replace CVaR1−ǫ

˜

X

by ess. sup( ˜

X). Suppose ˜ ζ has finite support {(Aζ, bζ)}ζ∈[N].

Theorem (Inner Approximation)

Z ⊇ ZS =

x ∈ Rn : Aζx ≥ bζ + δ

ǫ(x, −1)∗e, ∀ζ ∈ [N]

Remarks.

◮ ZS is a conic set ◮ ZS ≡ the robust scenario approach (Calafiore and Campi, 2006) to

regular CCP when the sample size is small

◮ ZS can be improved by other less conservative approximations

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 12 / 27

SLIDE 30

Inner Approximation: Worst-case CVaR

Note

f(x, ζ) := max
min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
≥ min

i∈[m]

(aζ

i )⊤x − bζ i

:=

g(x, ζ), Replace f(x, ζ) by g(x, ζ) and by monotonicity of coherent risk measure.

Theorem (Inner Approximation)

Z =

x : δ

ǫ(x, −1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
where

f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 13 / 27

SLIDE 31

Inner Approximation: Worst-case CVaR

Note

f(x, ζ) := max
min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
≥ min

i∈[m]

(aζ

i )⊤x − bζ i

:=

g(x, ζ), Replace f(x, ζ) by g(x, ζ) and by monotonicity of coherent risk measure.

Theorem (Inner Approximation)

Z ⊇ ZC =

x : δ

ǫ(x, −1)∗ + CVaR1−ǫ

−

g(x, ˜ ζ)

≤ 0
where

g(x, ζ) := min

i∈[m]

(aζ

i )⊤x − bζ i

.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 13 / 27

SLIDE 32

Inner Approximation: Worst-case CVaR

Note

f(x, ζ) := max
min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
≥ min

i∈[m]

(aζ

i )⊤x − bζ i

:=

g(x, ζ), Replace f(x, ζ) by g(x, ζ) and by monotonicity of coherent risk measure.

Theorem (Inner Approximation)

Z ⊇ ZC =

x : δ

ǫ(x, −1)∗ + CVaR1−ǫ

−

g(x, ˜ ζ)

≤ 0
where

g(x, ζ) := min

i∈[m]

(aζ

i )⊤x − bζ i

.

Remarks.

◮ ZC is a conic set ◮ ZC ≡ the worst-case CVaR approximation of DRCCP-W (Nemirovski

and Shapiro, 2006)

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 13 / 27

SLIDE 33

Illustrations of Z, ZVaR, ZC, ZS

Theorem (Model Comparison)

ZS ⊆ ZC ⊆ Z ⊆ ZVaR. x1 x2 (2, 2) (3, 2) (2, 3) ZVaR Consider

Z =

x :

inf

P∈PW P

˜

a1 ≤ x1 ˜ a2 ≤ x2

≥ 1 − ǫ
.

◮ Risk parameter ǫ = 2/3 ◮ Wasserstein radius δ = 1/6 ◮ N = 3 empirical data points:

(a1

1, a1 2) = (1, 3)

(a2

1, a2 2) = (3, 1)

(a3

1, a3 2) = (2, 2)

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 14 / 27

SLIDE 34

Illustrations of Z, ZVaR, ZC, ZS

Theorem (Model Comparison)

ZS ⊆ ZC ⊆ Z ⊆ ZVaR. x1 x2 (2, 2) (3, 2) (2, 3) ZVaR Z Consider

Z =

x :

inf

P∈PW P

˜

a1 ≤ x1 ˜ a2 ≤ x2

≥ 1 − ǫ
.

◮ Risk parameter ǫ = 2/3 ◮ Wasserstein radius δ = 1/6 ◮ N = 3 empirical data points:

(a1

1, a1 2) = (1, 3)

(a2

1, a2 2) = (3, 1)

(a3

1, a3 2) = (2, 2)

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 14 / 27

SLIDE 35

Illustrations of Z, ZVaR, ZC, ZS

Theorem (Model Comparison)

ZS ⊆ ZC ⊆ Z ⊆ ZVaR. x1 x2 (2, 2) (3, 2) (2, 3) ZVaR Z ZC Consider

Z =

x :

inf

P∈PW P

˜

a1 ≤ x1 ˜ a2 ≤ x2

≥ 1 − ǫ
.

◮ Risk parameter ǫ = 2/3 ◮ Wasserstein radius δ = 1/6 ◮ N = 3 empirical data points:

(a1

1, a1 2) = (1, 3)

(a2

1, a2 2) = (3, 1)

(a3

1, a3 2) = (2, 2)

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 14 / 27

SLIDE 36

Illustrations of Z, ZVaR, ZC, ZS

Theorem (Model Comparison)

ZS ⊆ ZC ⊆ Z ⊆ ZVaR. x1 x2 (2, 2) (3, 2) (2, 3) ZVaR Z ZC ZS Consider

Z =

x :

inf

P∈PW P

˜

a1 ≤ x1 ˜ a2 ≤ x2

≥ 1 − ǫ
.

◮ Risk parameter ǫ = 2/3 ◮ Wasserstein radius δ = 1/6 ◮ N = 3 empirical data points:

(a1

1, a1 2) = (1, 3)

(a2

1, a2 2) = (3, 1)

(a3

1, a3 2) = (2, 2)

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 14 / 27

SLIDE 37

Mixed Integer Program Reformulation

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 15 / 27

SLIDE 38

CVaR Reformulation: Recall

DRCCP-W set Z =

x : inf

P∈PW P

˜ Ax ≥ ˜ b

≥ 1 − ǫ
.

Theorem (Exact Formulation)

Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 16 / 27

SLIDE 39

Mixed Integer Program (MIP) Reformulation

DRCCP-W set Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 17 / 27

SLIDE 40

Mixed Integer Program (MIP) Reformulation

DRCCP-W set Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

◮ Linearize outer maximum with binary variable zζ ∈ {0, 1} and

continuous variable wζ as f(x, ζ) = wζ and wζ =      0, if min

i∈[m]

(aζ

i )⊤x − bζ i

≤ 0

min

i∈[m]

(aζ

i )⊤x − bζ i

,
therwise

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 17 / 27

SLIDE 41

Mixed Integer Program (MIP) Reformulation

DRCCP-W set Z =

x : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := max

min

i∈[m]

(aζ

i )⊤x − bζ i

, 0
.

◮ Linearize outer maximum with binary variable zζ ∈ {0, 1} and

continuous variable wζ as f(x, ζ) = wζ and 0 ≤ wζ ≤ Mζzζ wζ − Mζ(1 − zζ) ≤ (aζ

i )⊤x − bζ i , ∀i ∈ [m]

where Mζ ≥ max

x∈Z min i∈[m]

(aζ

i )⊤x − bζ i

Xie (Virginia Tech)

DRCCP with Wasserstein Distance June 26, 2019 17 / 27

SLIDE 42

Mixed Integer Program (MIP) Reformulation

DRCCP-W set Z =            x : δ ǫ(x, 1)∗ + CVaR1−ǫ

−wζ

≤ 0 0 ≤ wζ ≤ Mζzζ, ∀ζ wζ − Mζ(1 − zζ) ≤ (aζ

i )⊤x − bζ i , ∀i ∈ [m], ∀ζ

zζ ∈ {0, 1}, ∀ζ           

◮ Empirical distribution is finite-support {(Aζ, bζ)}ζ∈[N] ⇒ set Z is an

MIP

◮ Optimality is guaranteed by the solvers

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 17 / 27

SLIDE 43

Mixed Integer Program (MIP) Reformulation

DRCCP-W set Z =            x : δ ǫ(x, 1)∗ + CVaR1−ǫ

−wζ

≤ 0 0 ≤ wζ ≤ Mζzζ, ∀ζ wζ − Mζ(1 − zζ) ≤ (aζ

i )⊤x − bζ i , ∀i ∈ [m], ∀ζ

zζ ∈ {0, 1}, ∀ζ           

◮ Empirical distribution is finite-support {(Aζ, bζ)}ζ∈[N] ⇒ set Z is an

MIP

◮ Optimality is guaranteed by the solvers ◮ Similar to regular CCP, (1) big M coefficients weaken the formulation,

(2) number of binary variables grows as sample size N increases

Both will be addressed for binary DRCCP Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 17 / 27

SLIDE 44

Binary DRCCP-W and Submodularity

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SLIDE 45

Preliminaries

Binary DRCCP-W set Z =

x ∈ {0, 1}n : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

max
(aζ

i )⊤x − bζ i , 0

.

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SLIDE 46

Preliminaries

Binary DRCCP-W set Z =

x ∈ {0, 1}n : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

max
(aζ

i )⊤x − bζ i , 0

.

Fact 1

Given d1 ∈ Rn

+, d2, d3 ∈ R, function f(x) = − max

d⊤

1 x + d2, d3

is

submodular over the binary hypercube.

Fact 2 (Edmonds, 1970)

For a submodular function f : {0, 1}n → R, conv(epi(f)) = conv {(x, w) : f(x) ≤ w, x ∈ {0, 1}n} = “extended polymatroid inequalities” (EPI) The time complexity of separation over EPI is O(n log(n))

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SLIDE 47

Binary DRCCP-W: Submodular Constrained Reformulation

Binary DRCCP-W set Z =

x ∈ {0, 1}n : δ

ǫ(x, 1)∗ + CVaR1−ǫ

−

f(x, ˜ ζ)

≤ 0
,

where f(x, ζ) := min

i∈[m]

max
(aζ

i )⊤x − bζ i , 0

.

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 20 / 27

SLIDE 48

Binary DRCCP-W: Submodular Constrained Reformulation

Binary DRCCP-W set Z =   x ∈ {0, 1}n : δ ǫ(x, 1)∗ + CVaR1−ǫ

w

˜ ζ

≤ 0 − max

(aζ

i )⊤x − bζ i , 0

≤ wζ, ∀i ∈ [m], ∀ζ

  

◮ Let wζ =

f(x, ζ) and linearize it

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 20 / 27

SLIDE 49

Binary DRCCP-W: Submodular Constrained Reformulation

Binary DRCCP-W set Z =              x : δ ǫ(x, 1)∗ + CVaR1−ǫ

w

˜ ζ

≤ 0 − max

(

aζ,x

i

)⊤x + ( aζ,y

i )⊤y −

bζ

i , 0

≤ wζ , ∀i ∈ [m], ∀ζ

xr + yr = 1, ∀r ∈ [n], x, y ∈ {0, 1}n             

◮ Let wζ =

f(x, ζ) and linearize it

◮ Let yr = 1 − xr and choose vectors

aζ,x

i

, aζ,y

i

∈ Rn

+ such that

(aζ

i )⊤x − bζ i = (

aζ,x

i

)⊤x + ( aζ,y

i )⊤y −

bζ

i

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 20 / 27

SLIDE 50

Binary DRCCP-W: Submodular Constrained Reformulation

Binary DRCCP-W set Z =              x : δ ǫ(x, 1)∗ + CVaR1−ǫ

w

˜ ζ

≤ 0 − max

(

aζ,x

i

)⊤x + ( aζ,y

i )⊤y −

bζ

i , 0

≤ wζ , ∀i ∈ [m], ∀ζ

xr + yr = 1, ∀r ∈ [n], x, y ∈ {0, 1}n             

◮ Let wζ =

f(x, ζ) and linearize it

◮ Let yr = 1 − xr and choose vectors

aζ,x

i

, aζ,y

i

∈ Rn

+ such that

(aζ

i )⊤x − bζ i = (

aζ,x

i

)⊤x + ( aζ,y

i )⊤y −

bζ

i

◮ Facts 1 and 2⇒(1) Z is submodular constrained set and

(2)separation of these constraints is very efficient

Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 20 / 27

SLIDE 51

Numerical Illustration : Setting

Consider distributionally robust chance constrained knapsack problem v∗ = max

x

c⊤x, s.t. inf

P∈P P

˜

a⊤

i x ≤ ˜

bi, ∀i ∈ [m]

≥ 1 − ǫ.

◮ Let n = 20, m = 10 ◮ Generate 10 random instances and for each instance, there are N = 100

samples.

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SLIDE 52

Results (1): Continuous Knapsack x ∈ [0, 1]n

ǫ δ Instances BigM Model VaR Model CVaR Model Opt.Val Time Value GAP Time Value GAP Time 0.05 0.01 1 54.93 6.11 56.37 2.62% 3.37 54.30 1.14% 0.06 2 47.69 5.24 48.79 2.29% 2.04 47.16 1.11% 0.05 3 50.73 4.44 51.43 1.38% 4.43 50.38 0.70% 0.05 4 53.97 3.61 54.98 1.87% 4.75 52.72 2.32% 0.06 5 54.96 6.99 56.44 2.68% 4.20 52.88 3.79% 0.05 6 56.03 6.46 57.40 2.44% 2.64 54.97 1.89% 0.05 7 54.17 6.69 55.04 1.62% 3.68 53.26 1.67% 0.05 8 55.40 5.81 56.55 2.09% 3.19 54.15 2.26% 0.05 9 57.63 4.91 58.95 2.29% 4.20 57.07 0.96% 0.05 10 56.31 4.34 57.15 1.50% 4.71 55.95 0.63% 0.06 Average 5.46 2.08% 3.72 1.65% 0.05 0.05 0.02 1 53.97 3.94 55.92 3.63% 3.27 53.83 0.24% 0.05 2 47.05 3.63 48.42 2.92% 3.20 46.79 0.53% 0.04 3 50.12 5.26 51.02 1.79% 4.48 49.96 0.33% 0.05 4 52.98 5.14 54.49 2.84% 4.83 52.28 1.33% 0.06 5 54.10 3.76 55.95 3.41% 3.67 52.44 3.07% 0.05 6 55.16 6.02 56.90 3.16% 3.33 54.52 1.17% 0.05 7 53.41 3.91 54.55 2.13% 3.81 52.83 1.08% 0.05 8 54.47 2.77 56.09 2.98% 3.34 53.71 1.39% 0.06 9 56.85 3.40 58.44 2.79% 4.00 56.59 0.46% 0.05 10 55.65 5.47 56.71 1.90% 4.90 55.53 0.22% 0.06 Average 4.33 2.76% 3.88 0.98% 0.05

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SLIDE 53

Results (2): Testing Robustness

Instances DRCCP Model CCP Model Target Violation (ǫ) δ∗ Opt.Val 90-Percentile Violation Opt.Val 90-Percentile Violation 1 0.03 53.76 0.042 56.99 0.135 0.05 2 0.02 50.06 0.044 52.67 0.087 3 0.03 52.37 0.031 55.11 0.153 4 0.01 56.94 0.039 58.33 0.096 5 0.02 53.38 0.028 55.89 0.121 6 0.02 50.25 0.032 52.13 0.096 7 0.01 59.38 0.047 60.98 0.080 8 0.03 54.60 0.047 57.77 0.129 9 0.03 62.51 0.047 66.39 0.118 10 0.03 52.82 0.036 56.90 0.132

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SLIDE 54

Results (3): Binary Knapsack x ∈ {0, 1}n

ǫ δ Instances n I MIP Formulation Submodular Formulation UB LB Time GAP

Opt. Val.

Time 0.05 0.1 1 20 10 93 86 3600.0 7.5% 89 49.3 2 20 10 97 90 3600.0 7.2% 95 30.6 3 20 10 95 84 3600.0 11.6% 90 387.0 4 20 10 84 74 3600.0 11.9% 78 275.7 5 20 10 87 81 3600.0 6.9% 82 140.4 6 20 10 97 85 3600.0 12.4% 88 972.5 7 20 10 89 75 3600.0 15.7% 84 169.6 8 20 10 100 88 3600.0 12.0% 96 80.5 9 20 10 96 78 3600.0 18.8% 92 59.3 10 20 10 93 93 3542.7 0.0% 93 18.2 Average 3594.3 10.4% 218.3 0.1 0.1 1 20 10 100 NA 3600.0 NA 92 172.9 2 20 10 106 NA 3600.0 NA 99 164.0 3 20 10 105 87 3600.0 17.1% 93 569.1 4 20 10 92 67 3600.0 27.2% 82 600.5 5 20 10 95 NA 3600.0 NA 86 332.0 6 20 10 109 NA 3600.0 NA 94 1852.4 7 20 10 96 NA 3600.0 NA 88 279.8 8 20 10 108 82 3600.0 24.1% 100 133.2 9 20 10 102 NA 3600.0 NA 94 389.3 10 20 10 103 96 3600.0 6.8% 96 149.7 Average 3600.0 18.8% 464.3

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SLIDE 55

Concluding Remarks

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SLIDE 56

Concluding Remarks

◮ DRCCP-W admits a CVaR interpretation

Derive inner and outer approximations

◮ DRCCP-W is mixed integer program representable

With big-M coefficients and additional binary variables

◮ Binary DRCCP-W ≡ a submodular constrained optimization problem

Without big-M coefficients or additional binary variables

References:

◮ W. Xie. “On Distributionally Robust Chance Constrained Program with Wasserstein

Distance”. Available at Optimization Online, 2018.

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SLIDE 57

Thank you!

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